Lie algebroids, gauge theories, and compatible geometrical structures

TL;DR

This paper explores the mathematical structures of Lie algebroids and their compatibility conditions with geometrical structures like metrics and symplectic forms, extending gauge theory concepts beyond traditional Lie groups.

Contribution

It establishes the Cartan condition as the compatibility criterion for connections on Lie algebroids and shows that such compatibility leads to Riemannian foliations and compatible structures on the base.

Findings

Compatibility of a connection with a Lie algebroid is the Cartan condition.

Gauge theory compatibility implies the foliation is Riemannian.

Proper Lie groupoid integrations admit compatible Riemannian structures.

Abstract

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the (possibly singular) foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove furthermore that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized metric structure.